
Here, \(x\)  is the physical distance between two nearby black holes, \(f_{\mathrm {PBH}}\)  is the fraction of PBHs in the dark matter. \(\bar{x}\)  is the mean separation of PBHs at matter-radiation equality, which is determined by PBH mass \(M_{\mathrm {BH}}\)  and energy density \(\rho _{\mathrm {BH}}(z_{\mathrm {eq}})\)  as \(\bar{x} = (M_{\mathrm {BH}}/\rho _{\mathrm {BH}}(z_{\mathrm {eq}}))^{1/3}\) . After interacting with the third black hole, the probability distribution function of binary system with respect to the major axes \(a\)  and eccentricity \(e\)  [1]} is
\(P(a, e) = \frac{3}{4} \frac{f^{3/2}_{\mathrm {PBH}} a^{1/2}}{\bar{x}^{3/2}} \frac{e}{(1-e^2)^{3/2}}~.\) 
